The tiered Aubry set for autonomous Lagrangian functions

If L is a Tonelli Lagrangian defined on the tangent bundle of a compact and connected manifold whose dimension is at least 2, we associate to L the tiered Aubry set and the tiered Mane set (defined in the article). We prove that the tiered Mane set is closed, connected, chain transitive and that if L is generic in the Mane sense, the tiered Mane set has no interior. Then, we give an example of such an explicit generic Tonelli Lagrangian function and an example proving that when M is the torus, the closure of the tiered Aubry set and the closure of the union of the K.A.M. tori may be different.Comment: 28 pages; to appear in Ann. Inst. Fourier number 58 (2008

Birational cobordism invariance of uniruled symplectic manifolds

A symplectic manifold $(M,\omega)$ is called {\em (symplectically) uniruled} if there is a nonzero genus zero GW invariant involving a point constraint. We prove that symplectic uniruledness is invariant under symplectic blow-up and blow-down. This theorem follows from a general Relative/Absolute correspondence for a symplectic manifold together with a symplectic submanifold. A direct consequence is that symplectic uniruledness is a symplectic birational invariant. Here we use Guillemin and Sternberg's notion of cobordism as the symplectic analogue of the birational equivalence.Comment: To appear in Invent. Mat

Higher-Order Triangular-Distance Delaunay Graphs: Graph-Theoretical Properties

We consider an extension of the triangular-distance Delaunay graphs (TD-Delaunay) on a set $P$ of points in the plane. In TD-Delaunay, the convex distance is defined by a fixed-oriented equilateral triangle $\triangledown$, and there is an edge between two points in $P$ if and only if there is an empty homothet of $\triangledown$ having the two points on its boundary. We consider higher-order triangular-distance Delaunay graphs, namely $k$-TD, which contains an edge between two points if the interior of the homothet of $\triangledown$ having the two points on its boundary contains at most $k$ points of $P$. We consider the connectivity, Hamiltonicity and perfect-matching admissibility of $k$-TD. Finally we consider the problem of blocking the edges of $k$-TD.Comment: 20 page

Coisotropic rigidity and C^0-symplectic geometry

We prove that symplectic homeomorphisms, in the sense of the celebrated Gromov-Eliashberg Theorem, preserve coisotropic submanifolds and their characteristic foliations. This result generalizes the Gromov-Eliashberg Theorem and demonstrates that previous rigidity results (on Lagrangians by Laudenbach-Sikorav, and on characteristics of hypersurfaces by Opshtein) are manifestations of a single rigidity phenomenon. To prove the above, we establish a C^0-dynamical property of coisotropic submanifolds which generalizes a foundational theorem in C^0-Hamiltonian dynamics: Uniqueness of generators for continuous analogs of Hamiltonian flows.Comment: 27 pages. v2. Significant reorganization of the paper, several typos and inaccuracies corrected after the refeering process. A theorem (Theorem 5, completing the study of C^0 dynamical properties of coisotropics) added. To appear in Duke Mathematical Journa

Harmonic analysis on Cayley Trees II: the Bose Einstein condensation

We investigate the Bose-Einstein Condensation on non homogeneous non amenable networks for the model describing arrays of Josephson junctions on perturbed Cayley Trees. The resulting topological model has also a mathematical interest in itself. The present paper is then the application to the Bose-Einstein Condensation phenomena, of the harmonic analysis aspects arising from additive and density zero perturbations, previously investigated by the author in a separate work. Concerning the appearance of the Bose-Einstein Condensation, the results are surprisingly in accordance with the previous ones, despite the lack of amenability. We indeed first show the following fact. Even when the critical density is finite (which is implied in all the models under consideration, thanks to the appearance of the hidden spectrum), if the adjacency operator of the graph is recurrent, it is impossible to exhibit temperature locally normal states (i.e. states for which the local particle density is finite) describing the condensation at all. The same occurs in the transient cases for which it is impossible to exhibit locally normal states describing the Bose--Einstein Condensation at mean particle density strictly greater than the critical density . In addition, for the transient cases, in order to construct locally normal temperature states through infinite volume limits of finite volume Gibbs states, a careful choice of the the sequence of the finite volume chemical potential should be done. For all such states, the condensate is essentially allocated on the base--point supporting the perturbation. This leads that the particle density always coincide with the critical one. It is shown that all such temperature states are Kubo-Martin-Schwinger states for a natural dynamics. The construction of such a dynamics, which is a very delicate issue, is also done.Comment: 28 pages, 6 figures, 1 tabl

A signature invariant for knotted Klein graphs

We define some signature invariants for a class of knotted trivalent graphs using branched covers. We relate them to classical signatures of knots and links. Finally, we explain how to compute these invariants through the example of Kinoshita's knotted theta graph.Comment: 23 pages, many figures. Comments welcome ! Historical inaccuracy fixe